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Chapter 4: Problem 57

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 3 \ln x-\frac{1}{3} \ln y $$

Short Answer

Expert verified
The simplified expression is \(\ln(\frac{x^3}{y^{\frac{1}{3}}})\)

Step by step solution

01

Apply the rule of exponents to both terms

The expression can be rewritten as \(\ln(x^3) - \ln(y^{1/3})\)
02

Apply the subtraction rule

The expression can be further simplified to \(\ln(\frac{x^3}{y^{1/3}})\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Properties of Logarithms
Logarithms have several useful properties that make them quite powerful in mathematical expressions. Two of the essential properties are the product rule and the power rule.

The product rule tells us that the logarithm of a product is the sum of the logarithms:
  • \( \log_b(mn) = \log_b(m) + \log_b(n) \)


The power rule helps us manage exponents conveniently:
  • \( \log_b(m^n) = n \cdot \log_b(m) \)


These properties allow us to simplify and manipulate logarithmic expressions, making our calculations more manageable.
Condensing Logarithmic Expressions
Condensing logarithms involves combining multiple logarithmic terms into a single term. This process uses the properties of logarithms to rewrite expressions compactly.

For example, you might encounter expressions like \( \ln a + \ln b \) and condense them to \( \ln(ab) \). The goal is to transform expressions using the rules systematically, resulting in a more simplified form.

The condensing process is essential in solving equations where working with a single logarithmic term is easier and more efficient.
Exponential Rules
The process of converting expressions involves understanding the rules of exponents, especially when working with logarithms.

In expressions like \( 3 \ln x \), you can rewrite them using the power rule:
  • \( \ln(x^3) \)


Similarly, \( \frac{1}{3} \ln y \) can be rewritten as \( \ln(y^{1/3}) \).

These transformations help in rewriting log expressions as one single term, simplifying further operations such as subtraction, as we've seen in this exercise.
Subtraction Rule in Logarithms
The subtraction rule in logarithms helps us understand the difference between two logarithmic expressions. This rule says:
  • \( \log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right) \)

Using this rule, if you have an expression like \( \ln(x^3) - \ln(y^{1/3}) \), it becomes \( \ln\left(\frac{x^3}{y^{1/3}}\right) \), successfully condensing the terms.

This rule is handy for simplifying expressions and solving equations that involve differences between logs, making it an essential aspect of logarithmic calculations.

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Most popular questions from this chapter

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{9}(9 x) $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(\frac{x^{2} y}{z^{2}}\right) $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{2} \sqrt[5]{\frac{x y^{4}}{16}} $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{5}\left(\frac{125}{y}\right) $$

The loudness level of a sound can be expressed by comparing the sound's intensity to the intensity of a sound barely audible to the human car. The formula $$ D=10\left(\log I-\log I_{0}\right) $$ describes the loudness level of a sound, \(D\), in decibels, where \(I\) is the intensity of the sound, in watts per meter". and \(I_{0}\) is the intensity of a sound barely audible to the human ear. a. Express the formula so that the expression in parentheses is written as a single logarithm. b. Use the form of the formula from part (a) to answer this question: If a sound has an intensity 100 times the intensity of a softer sound, how much larger on the decibel scale is the loudness level of the more intense sound?
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